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Confluence Proofs of Lambda-Mu-Calculi by Z Theorem

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Abstract

This paper applies Dehornoy et al.’s Z theorem and its variant, called the compositional Z theorem, to prove confluence of Parigot’s \(\lambda \mu \)-calculi extended by the simplification rules. First, it is proved that Baba et al.’s modified complete developments for the call-by-name and the call-by-value variants of the \(\lambda \mu \)-calculus with the renaming rule, which is one of the simplification rules, satisfy the Z property. It gives new confluence proofs for them by the Z theorem. Secondly, it is shown that the compositional Z theorem can be applied to prove confluence of the call-by-name and the call-by-value \(\lambda \mu \)-calculi with both simplification rules, the renaming and the \(\mu \eta \)-rules, whereas it is hard to apply the ordinary parallel reduction technique or the original Z theorem by one-pass definition of mappings for these variants.

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Acknowledgements

This work was partially supported by Grants-in-Aid for Scientific Research KAKENHI 18K11161 (to Yuki Honda and Koji Nakazawa) and 17K05343 and 20K03711 (to Ken-etsu Fujita).

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Correspondence to Yuki Honda.

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Honda, Y., Nakazawa, K. & Fujita, Ke. Confluence Proofs of Lambda-Mu-Calculi by Z Theorem. Stud Logica 109, 917–936 (2021). https://doi.org/10.1007/s11225-020-09931-0

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  • DOI: https://doi.org/10.1007/s11225-020-09931-0

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